A square sheet of paper has area $6 \text{ cm}^2$. The front is white and the back is black. When the sheet is folded so that point $A$ rests on the diagonal as shown, the visible black area is equal to the visible white area. How many centimeters is $A$ from its original position? Express your answer in simplest radical form.
Explanation: Let $x$ be the length of a leg of the black isosceles triangle.  Then the black area is $\frac{1}{2}(x)(x)=\frac{1}{2}x^2$.  The white area is $6-x^2$.  Solving $\frac{1}{2}x^2=6-x^2$, we find $x^2=4$, so $x=2$.  The distance from A to its original position is the length of a hypotenuse of a right triangle whose legs have length $x$.  Therefore, A is $\boxed{2\sqrt{2}}$ centimeters from its original position.